LMFDB - Embedded newform 512.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [512,2,Mod(1,512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.08834058349$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	512.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -2.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} +4.24264 q^{11} -6.00000 q^{13} +2.82843 q^{15} -4.24264 q^{19} -4.00000 q^{21} -8.48528 q^{23} -1.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -5.65685 q^{31} -6.00000 q^{33} -5.65685 q^{35} -6.00000 q^{37} +8.48528 q^{39} +6.00000 q^{41} -4.24264 q^{43} +2.00000 q^{45} +1.00000 q^{49} +2.00000 q^{53} -8.48528 q^{55} +6.00000 q^{57} +1.41421 q^{59} -6.00000 q^{61} -2.82843 q^{63} +12.0000 q^{65} +12.7279 q^{67} +12.0000 q^{69} +8.48528 q^{71} -12.0000 q^{73} +1.41421 q^{75} +12.0000 q^{77} +5.65685 q^{79} -5.00000 q^{81} +4.24264 q^{83} +2.82843 q^{87} -12.0000 q^{89} -16.9706 q^{91} +8.00000 q^{93} +8.48528 q^{95} -8.00000 q^{97} -4.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{9} - 12 q^{13} - 8 q^{21} - 2 q^{25} - 4 q^{29} - 12 q^{33} - 12 q^{37} + 12 q^{41} + 4 q^{45} + 2 q^{49} + 4 q^{53} + 12 q^{57} - 12 q^{61} + 24 q^{65} + 24 q^{69} - 24 q^{73} + 24 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^9 - 12 * q^13 - 8 * q^21 - 2 * q^25 - 4 * q^29 - 12 * q^33 - 12 * q^37 + 12 * q^41 + 4 * q^45 + 2 * q^49 + 4 * q^53 + 12 * q^57 - 12 * q^61 + 24 * q^65 + 24 * q^69 - 24 * q^73 + 24 * q^77 - 10 * q^81 - 24 * q^89 + 16 * q^93 - 16 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−5.65685	−0.956183
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	8.48528	1.35873
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.24264	−0.646997	−0.323498	−	0.946229i	$-0.604859\pi$
$43$	−4.24264	−0.646997	−0.323498	+	0.946229i	$0.604859\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−8.48528	−1.14416
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	1.41421	0.184115	0.0920575	−	0.995754i	$-0.470656\pi$
$59$	1.41421	0.184115	0.0920575	+	0.995754i	$0.470656\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	12.7279	1.55496	0.777482	−	0.628906i	$-0.216497\pi$
$67$	12.7279	1.55496	0.777482	+	0.628906i	$0.216497\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	4.24264	0.465690	0.232845	−	0.972514i	$-0.425196\pi$
$83$	4.24264	0.465690	0.232845	+	0.972514i	$0.425196\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−16.9706	−1.77900
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	8.48528	0.870572
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−4.24264	−0.426401
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	−7.07107	−0.683586	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107	−0.683586	−0.341793	+	0.939775i	$0.611034\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	8.48528	0.805387
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	16.9706	1.58251
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	−8.48528	−0.765092
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	1.41421	0.123560	0.0617802	−	0.998090i	$-0.480322\pi$
$131$	1.41421	0.123560	0.0617802	+	0.998090i	$0.480322\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	−11.3137	−0.973729
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.24264	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$139$	−4.24264	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.4558	−2.12872
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.3137	0.908739
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	−2.82843	−0.224309
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	21.2132	1.66155	0.830773	−	0.556611i	$-0.187899\pi$
$163$	21.2132	1.66155	0.830773	+	0.556611i	$0.187899\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	−8.48528	−0.656611	−0.328305	−	0.944572i	$-0.606478\pi$
$167$	−8.48528	−0.656611	−0.328305	+	0.944572i	$0.606478\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	4.24264	0.324443
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	−2.00000	−0.150329
$178$	0	0
$179$	−18.3848	−1.37414	−0.687071	−	0.726590i	$-0.741104\pi$
$179$	−18.3848	−1.37414	−0.687071	+	0.726590i	$0.741104\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	−16.9706	−1.21529
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−19.7990	−1.40351	−0.701757	−	0.712417i	$-0.747601\pi$
$199$	−19.7990	−1.40351	−0.701757	+	0.712417i	$0.747601\pi$
$200$	0	0
$201$	−18.0000	−1.26962
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	8.48528	0.589768
$208$	0	0
$209$	−18.0000	−1.24509
$210$	0	0
$211$	4.24264	0.292075	0.146038	−	0.989279i	$-0.453348\pi$
$211$	4.24264	0.292075	0.146038	+	0.989279i	$0.453348\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	8.48528	0.578691
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	16.9706	1.14676
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.7279	−0.844782	−0.422391	−	0.906414i	$-0.638809\pi$
$227$	−12.7279	−0.844782	−0.422391	+	0.906414i	$0.638809\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	−16.9706	−1.11658
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−16.9706	−1.09773	−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706	−1.09773	−0.548867	+	0.835910i	$0.684941\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	25.4558	1.61972
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	12.7279	0.803379	0.401690	−	0.915776i	$-0.368423\pi$
$251$	12.7279	0.803379	0.401690	+	0.915776i	$0.368423\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−16.9706	−1.05450
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	8.48528	0.523225	0.261612	−	0.965173i	$-0.415746\pi$
$263$	8.48528	0.523225	0.261612	+	0.965173i	$0.415746\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	16.9706	1.03858
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	22.6274	1.37452	0.687259	−	0.726413i	$-0.258814\pi$
$271$	22.6274	1.37452	0.687259	+	0.726413i	$0.258814\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	5.65685	0.338667
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−12.7279	−0.756596	−0.378298	−	0.925684i	$-0.623491\pi$
$283$	−12.7279	−0.756596	−0.378298	+	0.925684i	$0.623491\pi$
$284$	0	0
$285$	−12.0000	−0.710819
$286$	0	0
$287$	16.9706	1.00174
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	11.3137	0.663221
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	−2.82843	−0.164677
$296$	0	0
$297$	24.0000	1.39262
$298$	0	0
$299$	50.9117	2.94430
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	−14.1421	−0.812444
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	−4.24264	−0.242140	−0.121070	−	0.992644i	$-0.538633\pi$
$307$	−4.24264	−0.242140	−0.121070	+	0.992644i	$0.538633\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	5.65685	0.318728
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−8.48528	−0.475085
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	−8.48528	−0.469237
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−29.6985	−1.63238	−0.816188	−	0.577786i	$-0.803917\pi$
$331$	−29.6985	−1.63238	−0.816188	+	0.577786i	$0.803917\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−25.4558	−1.39080
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−8.48528	−0.460857
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	−21.2132	−1.13878	−0.569392	−	0.822066i	$-0.692821\pi$
$347$	−21.2132	−1.13878	−0.569392	+	0.822066i	$0.692821\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−33.9411	−1.81164
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−16.9706	−0.900704
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	−9.89949	−0.519589
$364$	0	0
$365$	24.0000	1.25622
$366$	0	0
$367$	11.3137	0.590571	0.295285	−	0.955409i	$-0.404585\pi$
$367$	11.3137	0.590571	0.295285	+	0.955409i	$0.404585\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	−16.9706	−0.876356
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	12.7279	0.653789	0.326895	−	0.945061i	$-0.393998\pi$
$379$	12.7279	0.653789	0.326895	+	0.945061i	$0.393998\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	33.9411	1.73431	0.867155	−	0.498038i	$-0.165946\pi$
$383$	33.9411	1.73431	0.867155	+	0.498038i	$0.165946\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	4.24264	0.215666
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.00000	−0.100887
$394$	0	0
$395$	−11.3137	−0.569254
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	16.9706	0.849591
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	33.9411	1.69073
$404$	0	0
$405$	10.0000	0.496904
$406$	0	0
$407$	−25.4558	−1.26180
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	8.48528	0.418548
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−8.48528	−0.416526
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	12.7279	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$419$	12.7279	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	36.0000	1.73810
$430$	0	0
$431$	16.9706	0.817443	0.408722	−	0.912659i	$-0.365975\pi$
$431$	16.9706	0.817443	0.408722	+	0.912659i	$0.365975\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	−5.65685	−0.271225
$436$	0	0
$437$	36.0000	1.72211
$438$	0	0
$439$	−2.82843	−0.134993	−0.0674967	−	0.997719i	$-0.521501\pi$
$439$	−2.82843	−0.134993	−0.0674967	+	0.997719i	$0.521501\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−12.7279	−0.604722	−0.302361	−	0.953194i	$-0.597775\pi$
$443$	−12.7279	−0.604722	−0.302361	+	0.953194i	$0.597775\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	−31.1127	−1.47158
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	25.4558	1.19867
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	33.9411	1.59118
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	−5.65685	−0.262896	−0.131448	−	0.991323i	$-0.541963\pi$
$463$	−5.65685	−0.262896	−0.131448	+	0.991323i	$0.541963\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	−4.24264	−0.196326	−0.0981630	−	0.995170i	$-0.531297\pi$
$467$	−4.24264	−0.196326	−0.0981630	+	0.995170i	$0.531297\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	8.48528	0.390981
$472$	0	0
$473$	−18.0000	−0.827641
$474$	0	0
$475$	4.24264	0.194666
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	16.9706	0.775405	0.387702	−	0.921785i	$-0.373269\pi$
$479$	16.9706	0.775405	0.387702	+	0.921785i	$0.373269\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	33.9411	1.54437
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	36.7696	1.66619	0.833094	−	0.553132i	$-0.186567\pi$
$487$	36.7696	1.66619	0.833094	+	0.553132i	$0.186567\pi$
$488$	0	0
$489$	−30.0000	−1.35665
$490$	0	0
$491$	−26.8701	−1.21263	−0.606314	−	0.795225i	$-0.707353\pi$
$491$	−26.8701	−1.21263	−0.606314	+	0.795225i	$0.707353\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	8.48528	0.381385
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	4.24264	0.189927	0.0949633	−	0.995481i	$-0.469727\pi$
$499$	4.24264	0.189927	0.0949633	+	0.995481i	$0.469727\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−42.4264	−1.89170	−0.945850	−	0.324604i	$-0.894769\pi$
$503$	−42.4264	−1.89170	−0.945850	+	0.324604i	$0.894769\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	−32.5269	−1.44457
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	−33.9411	−1.50147
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	5.65685	0.249271
$516$	0	0
$517$	0	0
$518$	0	0
$519$	19.7990	0.869079
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	4.24264	0.185518	0.0927589	−	0.995689i	$-0.470431\pi$
$523$	4.24264	0.185518	0.0927589	+	0.995689i	$0.470431\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	0	0
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	−1.41421	−0.0613716
$532$	0	0
$533$	−36.0000	−1.55933
$534$	0	0
$535$	14.1421	0.611418
$536$	0	0
$537$	26.0000	1.12198
$538$	0	0
$539$	4.24264	0.182743
$540$	0	0
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	0	0
$543$	25.4558	1.09241
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−21.2132	−0.907011	−0.453506	−	0.891253i	$-0.649827\pi$
$547$	−21.2132	−0.907011	−0.453506	+	0.891253i	$0.649827\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	8.48528	0.361485
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−16.9706	−0.720360
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	25.4558	1.07667
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.6985	1.25164	0.625821	−	0.779967i	$-0.284764\pi$
$563$	29.6985	1.25164	0.625821	+	0.779967i	$0.284764\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	−14.1421	−0.593914
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−12.7279	−0.532647	−0.266323	−	0.963884i	$-0.585809\pi$
$571$	−12.7279	−0.532647	−0.266323	+	0.963884i	$0.585809\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−33.9411	−1.41055
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	8.48528	0.351424
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	15.5563	0.642079	0.321040	−	0.947066i	$-0.395968\pi$
$587$	15.5563	0.642079	0.321040	+	0.947066i	$0.395968\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	14.1421	0.581730
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.0000	1.14596
$598$	0	0
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	−12.7279	−0.518321
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−22.6274	−0.918419	−0.459209	−	0.888328i	$-0.651867\pi$
$607$	−22.6274	−0.918419	−0.459209	+	0.888328i	$0.651867\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	0	0
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	16.9706	0.684319
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	38.1838	1.53474	0.767368	−	0.641207i	$-0.221566\pi$
$619$	38.1838	1.53474	0.767368	+	0.641207i	$0.221566\pi$
$620$	0	0
$621$	−48.0000	−1.92617
$622$	0	0
$623$	−33.9411	−1.35982
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	25.4558	1.01661
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.1127	−1.23858	−0.619288	−	0.785164i	$-0.712579\pi$
$631$	−31.1127	−1.23858	−0.619288	+	0.785164i	$0.712579\pi$
$632$	0	0
$633$	−6.00000	−0.238479
$634$	0	0
$635$	11.3137	0.448971
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	−8.48528	−0.335673
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	12.7279	0.501940	0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279	0.501940	0.250970	+	0.967995i	$0.419250\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−25.4558	−1.00077	−0.500386	−	0.865802i	$-0.666809\pi$
$647$	−25.4558	−1.00077	−0.500386	+	0.865802i	$0.666809\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	22.6274	0.886838
$652$	0	0
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	0	0
$655$	−2.82843	−0.110516
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	−41.0122	−1.59761	−0.798804	−	0.601591i	$-0.794534\pi$
$659$	−41.0122	−1.59761	−0.798804	+	0.601591i	$0.794534\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	16.9706	0.657103
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−25.4558	−0.982712
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	−22.6274	−0.868361
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	−38.1838	−1.46106	−0.730531	−	0.682880i	$-0.760727\pi$
$683$	−38.1838	−1.46106	−0.730531	+	0.682880i	$0.760727\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	25.4558	0.971201
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−29.6985	−1.12978	−0.564892	−	0.825165i	$-0.691082\pi$
$691$	−29.6985	−1.12978	−0.564892	+	0.825165i	$0.691082\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.9706	0.641886
$700$	0	0
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	0	0
$703$	25.4558	0.960085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.2843	1.06374
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	−5.65685	−0.212149
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	50.9117	1.90399
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−2.82843	−0.104901	−0.0524503	−	0.998624i	$-0.516703\pi$
$727$	−2.82843	−0.104901	−0.0524503	+	0.998624i	$0.516703\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	0	0
$735$	2.82843	0.104328
$736$	0	0
$737$	54.0000	1.98912
$738$	0	0
$739$	−46.6690	−1.71675	−0.858374	−	0.513024i	$-0.828525\pi$
$739$	−46.6690	−1.71675	−0.858374	+	0.513024i	$0.828525\pi$
$740$	0	0
$741$	−36.0000	−1.32249
$742$	0	0
$743$	25.4558	0.933884	0.466942	−	0.884288i	$-0.345356\pi$
$743$	25.4558	0.933884	0.466942	+	0.884288i	$0.345356\pi$
$744$	0	0
$745$	−44.0000	−1.61204
$746$	0	0
$747$	−4.24264	−0.155230
$748$	0	0
$749$	−20.0000	−0.730784
$750$	0	0
$751$	5.65685	0.206422	0.103211	−	0.994660i	$-0.467088\pi$
$751$	5.65685	0.206422	0.103211	+	0.994660i	$0.467088\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	−28.2843	−1.02937
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	50.9117	1.84798
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	16.9706	0.614376
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.48528	−0.306386
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	−8.48528	−0.305590
$772$	0	0
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	−25.4558	−0.912050
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−29.6985	−1.05864	−0.529318	−	0.848423i	$-0.677552\pi$
$787$	−29.6985	−1.05864	−0.529318	+	0.848423i	$0.677552\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	16.9706	0.603404
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	5.65685	0.200628
$796$	0	0
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−50.9117	−1.79663
$804$	0	0
$805$	48.0000	1.69178
$806$	0	0
$807$	−19.7990	−0.696957
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	4.24264	0.148979	0.0744896	−	0.997222i	$-0.476267\pi$
$811$	4.24264	0.148979	0.0744896	+	0.997222i	$0.476267\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	0	0
$815$	−42.4264	−1.48613
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	16.9706	0.592999
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	19.7990	0.690149	0.345075	−	0.938575i	$-0.387854\pi$
$823$	19.7990	0.690149	0.345075	+	0.938575i	$0.387854\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	43.8406	1.52449	0.762244	−	0.647290i	$-0.224098\pi$
$827$	43.8406	1.52449	0.762244	+	0.647290i	$0.224098\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	8.48528	0.294351
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.9706	0.587291
$836$	0	0
$837$	−32.0000	−1.10608
$838$	0	0
$839$	−8.48528	−0.292944	−0.146472	−	0.989215i	$-0.546792\pi$
$839$	−8.48528	−0.292944	−0.146472	+	0.989215i	$0.546792\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−16.9706	−0.584497
$844$	0	0
$845$	−46.0000	−1.58245
$846$	0	0
$847$	19.7990	0.680301
$848$	0	0
$849$	18.0000	0.617758
$850$	0	0
$851$	50.9117	1.74523
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	−8.48528	−0.290191
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	46.6690	1.59233	0.796164	−	0.605081i	$-0.206859\pi$
$859$	46.6690	1.59233	0.796164	+	0.605081i	$0.206859\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	50.9117	1.73305	0.866527	−	0.499130i	$-0.166347\pi$
$863$	50.9117	1.73305	0.866527	+	0.499130i	$0.166347\pi$
$864$	0	0
$865$	28.0000	0.952029
$866$	0	0
$867$	24.0416	0.816497
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−76.3675	−2.58762
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	33.9411	1.14742
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	14.1421	0.477002
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−38.1838	−1.28499	−0.642493	−	0.766292i	$-0.722100\pi$
$883$	−38.1838	−1.28499	−0.642493	+	0.766292i	$0.722100\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−21.2132	−0.710669
$892$	0	0
$893$	0	0
$894$	0	0
$895$	36.7696	1.22907
$896$	0	0
$897$	−72.0000	−2.40401
$898$	0	0
$899$	11.3137	0.377333
$900$	0	0
$901$	0	0
$902$	0	0
$903$	16.9706	0.564745
$904$	0	0
$905$	36.0000	1.19668
$906$	0	0
$907$	−4.24264	−0.140875	−0.0704373	−	0.997516i	$-0.522439\pi$
$907$	−4.24264	−0.140875	−0.0704373	+	0.997516i	$0.522439\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	33.9411	1.12452	0.562260	−	0.826961i	$-0.309932\pi$
$911$	33.9411	1.12452	0.562260	+	0.826961i	$0.309932\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	0	0
$915$	−16.9706	−0.561029
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	2.82843	0.0933012	0.0466506	−	0.998911i	$-0.485145\pi$
$919$	2.82843	0.0933012	0.0466506	+	0.998911i	$0.485145\pi$
$920$	0	0
$921$	6.00000	0.197707
$922$	0	0
$923$	−50.9117	−1.67578
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	2.82843	0.0928977
$928$	0	0
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	−4.24264	−0.139047
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	−8.48528	−0.276907
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	0	0
$943$	−50.9117	−1.65791
$944$	0	0
$945$	−32.0000	−1.04096
$946$	0	0
$947$	41.0122	1.33272	0.666359	−	0.745631i	$-0.267852\pi$
$947$	41.0122	1.33272	0.666359	+	0.745631i	$0.267852\pi$
$948$	0	0
$949$	72.0000	2.33722
$950$	0	0
$951$	14.1421	0.458590
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−33.9411	−1.09831
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	−16.9706	−0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	7.07107	0.227862
$964$	0	0
$965$	−48.0000	−1.54517
$966$	0	0
$967$	−31.1127	−1.00052	−0.500258	−	0.865876i	$-0.666762\pi$
$967$	−31.1127	−1.00052	−0.500258	+	0.865876i	$0.666762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.6985	0.953070	0.476535	−	0.879156i	$-0.341893\pi$
$971$	29.6985	0.953070	0.476535	+	0.879156i	$0.341893\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	−8.48528	−0.271746
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	−50.9117	−1.62714
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	42.4264	1.35319	0.676596	−	0.736354i	$-0.263454\pi$
$983$	42.4264	1.35319	0.676596	+	0.736354i	$0.263454\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	36.0000	1.14473
$990$	0	0
$991$	−45.2548	−1.43757	−0.718784	−	0.695234i	$-0.755301\pi$
$991$	−45.2548	−1.43757	−0.718784	+	0.695234i	$0.755301\pi$
$992$	0	0
$993$	42.0000	1.33283
$994$	0	0
$995$	39.5980	1.25534
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	512.2.a.b.1.1	✓	2
3.2	odd	2		4608.2.a.p.1.2		2
4.3	odd	2	inner	512.2.a.b.1.2	yes	2
8.3	odd	2		512.2.a.e.1.1	yes	2
8.5	even	2		512.2.a.e.1.2	yes	2
12.11	even	2		4608.2.a.p.1.1		2
16.3	odd	4		512.2.b.e.257.4		4
16.5	even	4		512.2.b.e.257.3		4
16.11	odd	4		512.2.b.e.257.1		4
16.13	even	4		512.2.b.e.257.2		4
24.5	odd	2		4608.2.a.c.1.2		2
24.11	even	2		4608.2.a.c.1.1		2
32.3	odd	8		1024.2.e.n.257.2		4
32.5	even	8		1024.2.e.n.769.1		4
32.11	odd	8		1024.2.e.n.769.2		4
32.13	even	8		1024.2.e.n.257.1		4
32.19	odd	8		1024.2.e.h.257.1		4
32.21	even	8		1024.2.e.h.769.2		4
32.27	odd	8		1024.2.e.h.769.1		4
32.29	even	8		1024.2.e.h.257.2		4
48.5	odd	4		4608.2.d.j.2305.3		4
48.11	even	4		4608.2.d.j.2305.4		4
48.29	odd	4		4608.2.d.j.2305.1		4
48.35	even	4		4608.2.d.j.2305.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.b.1.1	✓	2	1.1	even	1	trivial
512.2.a.b.1.2	yes	2	4.3	odd	2	inner
512.2.a.e.1.1	yes	2	8.3	odd	2
512.2.a.e.1.2	yes	2	8.5	even	2
512.2.b.e.257.1		4	16.11	odd	4
512.2.b.e.257.2		4	16.13	even	4
512.2.b.e.257.3		4	16.5	even	4
512.2.b.e.257.4		4	16.3	odd	4
1024.2.e.h.257.1		4	32.19	odd	8
1024.2.e.h.257.2		4	32.29	even	8
1024.2.e.h.769.1		4	32.27	odd	8
1024.2.e.h.769.2		4	32.21	even	8
1024.2.e.n.257.1		4	32.13	even	8
1024.2.e.n.257.2		4	32.3	odd	8
1024.2.e.n.769.1		4	32.5	even	8
1024.2.e.n.769.2		4	32.11	odd	8
4608.2.a.c.1.1		2	24.11	even	2
4608.2.a.c.1.2		2	24.5	odd	2
4608.2.a.p.1.1		2	12.11	even	2
4608.2.a.p.1.2		2	3.2	odd	2
4608.2.d.j.2305.1		4	48.29	odd	4
4608.2.d.j.2305.2		4	48.35	even	4
4608.2.d.j.2305.3		4	48.5	odd	4
4608.2.d.j.2305.4		4	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -2.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} +4.24264 q^{11} -6.00000 q^{13} +2.82843 q^{15} -4.24264 q^{19} -4.00000 q^{21} -8.48528 q^{23} -1.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -5.65685 q^{31} -6.00000 q^{33} -5.65685 q^{35} -6.00000 q^{37} +8.48528 q^{39} +6.00000 q^{41} -4.24264 q^{43} +2.00000 q^{45} +1.00000 q^{49} +2.00000 q^{53} -8.48528 q^{55} +6.00000 q^{57} +1.41421 q^{59} -6.00000 q^{61} -2.82843 q^{63} +12.0000 q^{65} +12.7279 q^{67} +12.0000 q^{69} +8.48528 q^{71} -12.0000 q^{73} +1.41421 q^{75} +12.0000 q^{77} +5.65685 q^{79} -5.00000 q^{81} +4.24264 q^{83} +2.82843 q^{87} -12.0000 q^{89} -16.9706 q^{91} +8.00000 q^{93} +8.48528 q^{95} -8.00000 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{9} - 12 q^{13} - 8 q^{21} - 2 q^{25} - 4 q^{29} - 12 q^{33} - 12 q^{37} + 12 q^{41} + 4 q^{45} + 2 q^{49} + 4 q^{53} + 12 q^{57} - 12 q^{61} + 24 q^{65} + 24 q^{69} - 24 q^{73} + 24 q^{77}+ \cdots - 16 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^9 - 12 * q^13 - 8 * q^21 - 2 * q^25 - 4 * q^29 - 12 * q^33 - 12 * q^37 + 12 * q^41 + 4 * q^45 + 2 * q^49 + 4 * q^53 + 12 * q^57 - 12 * q^61 + 24 * q^65 + 24 * q^69 - 24 * q^73 + 24 * q^77 - 10 * q^81 - 24 * q^89 + 16 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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